Your equation doesn't appear as it should on my browser, but there's a simple theorem for estimating the remainder [itex]R_N[/itex] that exists when the nth partial sum [itex]S_N[/itex] of a convergent series is computed. It says:

You're not computing the error, you're estimating it. And any series that converges by the integral test has only positive terms. So there is always an implicit lower boundary of 0 on [itex]R_N[/itex].